1. Quantum ShiftRegister Circuits (from a company in Northern Virginia) Mark M. Wilde arXiv:0903.3894 To appear in Physical Review A National Institute of Standards and Technology, Wednesday, June 10, 2009

2. Overview • Classical Shift Register Circuits • Examples with Classical CNOT gate • Quantum Shift Register Circuits • “Memory Consumption” Theorem • Future Work

3. Applications of Shift Registers Shift Registers and Convolutional Coding techniques have application in cellular and deep space communication ViterbiAlgorithmis most popular technique for determining errors

4. Compute output streams from memory bits Store input stream sequentially Classical Shift Registers (D represents “delay”)

5. Mathematical Representation Input stream is a binary sequence Output stream is a binary sequence Convolveinput stream with system function to get output stream: Can also represent input stream as a polynomial And same for output stream Multiplyinput with system function to get output polynomial:

6. Input: 1000000000000000 Input Polynomial: 1 Output: 1100000000000000 Output Polynomial: 1 + D Classical Shift Register Example

7. Input: 1000000000000000 Input Polynomial: 1 Output: 01111111111111111 Output Polynomial: D / (1 + D) Another Example

8. What is aquantumshift register? A quantum shift register circuit acts on a set of input qubits and memory qubits, outputs a set ofoutput qubits and updated memory qubits, and feeds the memory back into the device for the next cycle (similar to the operation of a classical shift register).

9. Quantum Circuit Depiction

10. Lattice Depiction

11. Unencoded Stabilizer Encoded Stabilizer Brief Intro to Stabilizer Formalism Laflamme et al., Physical Review Letters 77, 198-201 (1996).

12. Binary Vector Representation

13. CNOT Gate Binary Vector Transformation Pauli Operator Transformation

14. CNOT gate with Memory How to describe input, output, and memory?

15. Recursive Equations

16. D-Transform Input Vector Output Vector Transformation

17. CNOT gate with more memory Transformation

18. Combo Shift Register Circuits Is it possible to simplify?

19. Simplified Shift Register Circuit “Commute last gate through memory”

20. Example of a Code Check matrix of a CSSquantum convolutional code Use Grassl-Roetteler algorithm to decompose as CNOT(3,2)(1+1/D) CNOT(1,2)(D) CNOT(1,3)(1+D)

21. Quantum Shift Register Circuit

22. “CSS Shift Register Memory” Theorem Given a description of a quantum convolutional code, how large of a quantum memory do we need to implement? Proof uses induction and exhaustively considers all the ways that CNOT gates can combine

23. General Shift Register Circuit General technique applies to arbitrary quantum convolutional codes

24. Experimental Implementations? Optical lattices of neutral atoms Spin chains for state transfer Linear-optical circuits

25. Current Directions Extend Memory Consumption Theorem to arbitrary quantum convolutional codes Study the Entanglement Structure of states that are input to a quantum shift register circuit (Area Laws should apply here) THANK YOU!